Average Error: 14.9 → 0.5
Time: 3.5s
Precision: 64
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{1 + \left(\sqrt[3]{N + 1} \cdot \sqrt[3]{N + 1}\right) \cdot \left(\left(\sqrt[3]{N \cdot N - 1 \cdot 1} \cdot \frac{1}{\sqrt[3]{N - 1}}\right) \cdot N\right)}\]
\tan^{-1} \left(N + 1\right) - \tan^{-1} N
↓
\tan^{-1}_* \frac{1}{1 + \left(\sqrt[3]{N + 1} \cdot \sqrt[3]{N + 1}\right) \cdot \left(\left(\sqrt[3]{N \cdot N - 1 \cdot 1} \cdot \frac{1}{\sqrt[3]{N - 1}}\right) \cdot N\right)}
double f(double N) {
        double r137381 = N;
        double r137382 = 1.0;
        double r137383 = r137381 + r137382;
        double r137384 = atan(r137383);
        double r137385 = atan(r137381);
        double r137386 = r137384 - r137385;
        return r137386;
}
↓
double f(double N) {
        double r137387 = 1.0;
        double r137388 = 1.0;
        double r137389 = N;
        double r137390 = r137389 + r137387;
        double r137391 = cbrt(r137390);
        double r137392 = r137391 * r137391;
        double r137393 = r137389 * r137389;
        double r137394 = r137387 * r137387;
        double r137395 = r137393 - r137394;
        double r137396 = cbrt(r137395);
        double r137397 = r137389 - r137387;
        double r137398 = cbrt(r137397);
        double r137399 = r137388 / r137398;
        double r137400 = r137396 * r137399;
        double r137401 = r137400 * r137389;
        double r137402 = r137392 * r137401;
        double r137403 = r137388 + r137402;
        double r137404 = atan2(r137387, r137403);
        return r137404;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.9
Target0.3
Herbie0.5
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.9

    \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
  2. Using strategy rm

  3. Applied diff-atan13.8

    \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}}\]

  4. Simplified0.3

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{1}}{1 + \left(N + 1\right) \cdot N}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.6

    \[\leadsto \tan^{-1}_* \frac{1}{1 + \color{blue}{\left(\left(\sqrt[3]{N + 1} \cdot \sqrt[3]{N + 1}\right) \cdot \sqrt[3]{N + 1}\right)} \cdot N}\]

  7. Applied associate-*l*0.6

    \[\leadsto \tan^{-1}_* \frac{1}{1 + \color{blue}{\left(\sqrt[3]{N + 1} \cdot \sqrt[3]{N + 1}\right) \cdot \left(\sqrt[3]{N + 1} \cdot N\right)}}\]
  8. Using strategy rm

  9. Applied flip-+0.6

    \[\leadsto \tan^{-1}_* \frac{1}{1 + \left(\sqrt[3]{N + 1} \cdot \sqrt[3]{N + 1}\right) \cdot \left(\sqrt[3]{\color{blue}{\frac{N \cdot N - 1 \cdot 1}{N - 1}}} \cdot N\right)}\]

  10. Applied cbrt-div0.5

    \[\leadsto \tan^{-1}_* \frac{1}{1 + \left(\sqrt[3]{N + 1} \cdot \sqrt[3]{N + 1}\right) \cdot \left(\color{blue}{\frac{\sqrt[3]{N \cdot N - 1 \cdot 1}}{\sqrt[3]{N - 1}}} \cdot N\right)}\]
  11. Using strategy rm

  12. Applied div-inv0.5

    \[\leadsto \tan^{-1}_* \frac{1}{1 + \left(\sqrt[3]{N + 1} \cdot \sqrt[3]{N + 1}\right) \cdot \left(\color{blue}{\left(\sqrt[3]{N \cdot N - 1 \cdot 1} \cdot \frac{1}{\sqrt[3]{N - 1}}\right)} \cdot N\right)}\]

  13. Final simplification0.5

    \[\leadsto \tan^{-1}_* \frac{1}{1 + \left(\sqrt[3]{N + 1} \cdot \sqrt[3]{N + 1}\right) \cdot \left(\left(\sqrt[3]{N \cdot N - 1 \cdot 1} \cdot \frac{1}{\sqrt[3]{N - 1}}\right) \cdot N\right)}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (N)
  :name "2atan (example 3.5)"
  :precision binary64

  :herbie-target
  (atan (/ 1 (+ 1 (* N (+ N 1)))))

  (- (atan (+ N 1)) (atan N)))